= Solution
The operator $\hat J_z$ describes the $z$ component of <angular momentum>, whereas $\hat J^2$ describes its total squared magnitude. For a <spherical harmonic> $Y_{JM}$ the corresponding <eigenvalues> are
$$
\boxed{\hat J_zY_{JM}=M\hbar Y_{JM},\qquad
\hat J^2Y_{JM}=J(J+1)\hbar^2Y_{JM}.}
$$
Thus the magnitude is $\hbar\sqrt{J(J+1)}$, not $J\hbar$. For ordinary orbital rotation $J=0,1,2,\ldots$ and $M=-J,-J+1,\ldots,J$. The simultaneous values are possible because these two operators commute.
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